Lightly brush swordfish with olive oil, salt and pepper. Grill on medium heat for about 5 minutes per side (varies depending on thickness).

Slice mushrooms and put in a pan with 1/4 cup of broth. Cover and cook on medium heat for about 6 minutes or until soft. Drain. Pour mushrooms and broth in blender or food processor. Add remaining sauce ingredients except thyme. Blend until smooth. More broth might be needed to achieve desired consistency. Pour into pan and warm. Pour sauce over fish and sprinkle with thyme.